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Recurrence rates for shifts of finite type
Let ΣA be a topologically mixing shift of finite type, let σ : ΣA → ΣA be the usual left-shift, and let μ be the Gibbs measure for a H¨older continuous potential that is not cohomologous to a constant. In this paper we study recurrence rates for the dynamical system (ΣA, σ) that hold μ-almost surely...
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| Format: | Default Article |
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2024
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| Online Access: | https://hdl.handle.net/2134/27826533.v1 |
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| _version_ | 1818176527870722048 |
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| author | Demi Allen Simon Baker Barany Balazs |
| author_facet | Demi Allen Simon Baker Barany Balazs |
| author_sort | Demi Allen (14081889) |
| collection | Figshare |
| description | Let ΣA be a topologically mixing shift of finite type, let σ : ΣA → ΣA be the usual left-shift, and let μ be the Gibbs measure for a H¨older continuous potential that is not cohomologous to a constant. In this paper we study recurrence rates for the dynamical system (ΣA, σ) that hold μ-almost surely. In particular, given a function ψ : N → N we are interested in the following set Rψ = {i ∈ ΣA : in+1 . . . in+ψ(n)+1 = i1 . . . iψ(n) for infinitely many n ∈ N}. We provide sufficient conditions for μ(Rψ) = 1 and sufficient conditions for μ(Rψ) = 0. As a corollary of these results, we discover a new critical threshold where the measure of Rψ transitions from zero to one. This threshold was previously unknown even in the special case of a non-uniform Bernoulli measure defined on the full shift. The proofs of our results combine ideas from Probability Theory and Thermodynamic Formalism. In our final section we apply our results to the study of dynamics on self-similar sets. |
| format | Default Article |
| id | rr-article-27826533 |
| institution | Loughborough University |
| publishDate | 2024 |
| record_format | Figshare |
| spelling | rr-article-278265332024-11-22T14:58:52Z Recurrence rates for shifts of finite type Demi Allen (14081889) Simon Baker (13764742) Barany Balazs (20279517) Applied mathematics Mathematical physics Pure mathematics General Mathematics <p>Let Σ<sub>A</sub> be a topologically mixing shift of finite type, let σ : Σ<sub>A</sub> → Σ<sub>A</sub> be the usual left-shift, and let μ be the Gibbs measure for a H¨older continuous potential that is not cohomologous to a constant. In this paper we study recurrence rates for the dynamical system (Σ<sub>A</sub>, σ) that hold μ-almost surely. In particular, given a function ψ : N → N we are interested in the following set</p> <p>Rψ = {i ∈ ΣA : in+1 . . . in+ψ(n)+1 = i1 . . . iψ(n) for infinitely many n ∈ N}.</p> <p>We provide sufficient conditions for μ(Rψ) = 1 and sufficient conditions for μ(Rψ) = 0.</p> <p>As a corollary of these results, we discover a new critical threshold where the measure of Rψ transitions from zero to one. This threshold was previously unknown even in the special case of a non-uniform Bernoulli measure defined on the full shift. The proofs of our results combine ideas from Probability Theory and Thermodynamic Formalism. In our final section we apply our results to the study of dynamics on self-similar sets.</p> 2024-11-22T14:58:52Z Text Journal contribution 2134/27826533.v1 https://figshare.com/articles/journal_contribution/Recurrence_rates_for_shifts_of_finite_type/27826533 CC BY-NC-ND 4.0 |
| spellingShingle | Applied mathematics Mathematical physics Pure mathematics General Mathematics Demi Allen Simon Baker Barany Balazs Recurrence rates for shifts of finite type |
| title | Recurrence rates for shifts of finite type |
| title_full | Recurrence rates for shifts of finite type |
| title_fullStr | Recurrence rates for shifts of finite type |
| title_full_unstemmed | Recurrence rates for shifts of finite type |
| title_short | Recurrence rates for shifts of finite type |
| title_sort | recurrence rates for shifts of finite type |
| topic | Applied mathematics Mathematical physics Pure mathematics General Mathematics |
| url | https://hdl.handle.net/2134/27826533.v1 |